Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24285 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,20)(10,23)(11,15)(12,18)(13,21)(14,24)(17,30)(19,28)(22,34)(25,33)(26,32)(27,35)(29,31)(36,37);(6,8,10,18,11,19,24,33)(7,12,14,16,15,25,23,28)(9,17,21,31,22,32,35,37)(13,26,20,29,27,30,34,36);(12,25)(16,28)(26,30)(29,36);(6,9,10,21,11,22,24,35)(7,13,14,20,15,27,23,34)(8,17,18,31,19,32,33,37)(12,26,16,29,25,30,28,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17273);